Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the Lagrangian formalism in field theory.

I understand the motivation for it in classical mechanics and I do see the analogy between the classic Lagrangian and the Lagrangian density by taking the limit where you want to see the dynamics of an infinite number of particles (ie. each one being a point of a field) however I still can't realize the full picture as to why the Principle of Stationary Action should be valid for a field.

Is the Lagrangian formalism used simply because the equations were already known and people just constructed a Lagrangian that worked and provided the already known equations? Or was it built from the ground up already assuming the Principle of Stationary Action is valid ? If so why?

Note: I known a very similar question was already asked before here however I felt that the question was not answered fully.

EDIT: As I said I known that the question has been asked before however in the links provided and the links therein (which I had read prior to asking this question) an explicit answer to this is not provided.

$\begingroup$The normal modes of a field comprise an infinite collection of oscillators, so a field is simply a large mechanical system not qualitatively different than standard mechanics, of which analysis techniques can be suitably adapted. What makes you assume something has to go wrong?$\endgroup$
– Cosmas ZachosMar 2 at 18:34

$\begingroup$I guess it is the fact that in a rigid body I know that it is made up of "infinite" point particles which obey Newton's laws (which are equivalent to the principle of stationary action) however I don't think they apply to fields and as such I can't see the motivation to use the Lagrangian formalism to build field theory from the ground up.$\endgroup$
– ThunderSmotchMar 2 at 19:06

$\begingroup$Thank you for that source, I found the chapters 2 and 3 and the remarks therein made by the author have given me some footing as to why one should try to look for a Lagrangian field theory$\endgroup$
– ThunderSmotchMar 2 at 21:03